October  2017, 10(5): 1187-1206. doi: 10.3934/dcdss.2017065

Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: klcqu20132016@163.com (L. Kong)

Received  November 2016 Revised  January 2017 Published  June 2017

Fund Project: Acknowledgement: This work was supported by NSFC grant 11671058.

In the present paper the dynamics of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting is studied. We give out all the possible ranges of parameters for which the model has up to five equilibria. We prove that these equilibria can be topological saddles, nodes, foci, centers, saddle-nodes, cusps of codimension 2 or 3. Numerous kinds of bifurcations also occur, such as the transcritical bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Several numerical simulations are carried out to illustrate the validity of our results.

Citation: Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065
References:
[1]

A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program for Scientific Translations, Jerusalem-London, 1973. doi: 10.1016/0022-0396(77)90136-X. Google Scholar

[2]

R. I. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421. Google Scholar

[3]

F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8 (1979), 55-71. doi: 10.1007/BF00280586. Google Scholar

[4]

F. D. Chen, On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33-49. doi: 10.1016/j.cam.2004.10.001. Google Scholar

[5] S. N. ChowC. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, 1994. doi: 10.1017/CBO9780511665639. Google Scholar
[6] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. doi: 10.1016/0025-5564(87)90046-0. Google Scholar
[7]

C. W. Clark, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317-337. Google Scholar

[8] C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Re-sources, John Wiley & Sons, Inc., New York-London-Sydney, 1976. Google Scholar
[9]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity withnilponent linear part. The cusp case of codimension 3, Ergodic Theor. Dyn. Syst., 7 (1987), 375-413. doi: 10.1017/S0143385700004119. Google Scholar

[10]

T. C. Gard, Persistence in food webs: Holling-type food chains, Math. Biosci., 49 (1980), 61-67. doi: 10.1016/0025-5564(80)90110-8. Google Scholar

[11]

Y. Gong and J. Huang, Bogdanove-Takens bifurcations in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Apple. Sinica Eng. Ser., 30 (2014), 239-244. doi: 10.1007/s10255-014-0279-x. Google Scholar

[12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/2F978-1-4612-1140-2. Google Scholar
[13]

R. P. GuptaM. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366. doi: 10.1007/s12591-012-0142-6. Google Scholar

[14]

R. P. GuptaP. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear harvesting, Disc. Cont. Dyna. Sys. Ser. B., 20 (2015), 423-443. doi: 10.3934/dcdsb.2015.20.423. Google Scholar

[15]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math.Anal. Appl., 398 (2013), 278-295. doi: 10.1016/j.jmaa.2012.08.057. Google Scholar

[16]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5. Google Scholar

[17]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201. Google Scholar

[18]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Disc. Cont. Dyna. Sys. Ser. B., 18 (2013), 2101-2121. doi: 10.3934/dcdsb.2013.18.2101. Google Scholar

[19]

C. JostO. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey model, J. Comput. Bull. Math. Biol., 61 (1999), 19-32. doi: 10.1006/bulm.1998.0072. Google Scholar

[20]

S. V. KrishnaP. D. N. Srinivasu and B. Kaymackcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584. doi: 10.1006/bulm.1997.0023. Google Scholar

[21]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105. Google Scholar

[22] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2 edition, Springer-verlag, New York, 1998. doi: 10.1003/978-1-4757-3978-7. Google Scholar
[23]

Y. LamontagneC. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type Ⅲ functional response, J. Dynam. Diff. Equ., 20 (2008), 535-571. doi: 10.1007/s10884-008-9102-9. Google Scholar

[24]

K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Analysis: RAW., 12 (2011), 1961-1973. doi: 10.1016/j.nonrwa.2010.12.012. Google Scholar

[25]

K. Q. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys., 32 (2012), 901-933. doi: 10.3934/dcds.2012.32.901. Google Scholar

[26]

B. LeardC. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with non-constant harvesting, Disc. Cont. Dyna. Sys. Ser. S., 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. Google Scholar

[27]

P. Lenzini and J. Rebaza, Non-constant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci, 4 (2010), 791-803. Google Scholar

[28]

T. Lindstrom, Qualitative analysisnof a predator-prey system with limit cycles, J. Math. Biol., 31 (1993), 541-561. doi: 10.1007/BF00161198. Google Scholar

[29]

J. M. Lorca, E. G. Olivares and B. G. Yanez, The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, In: Mondaini, R. (ed. ) Proceedings of the International Symposium on Mathematical and Computational Biology: BIOMAT 2006, E-papers Servicos Editoriais Ltda. , R'io de Janeiro, (2007), 105–132.Google Scholar

[30] A. Lotka, Elements of Mathematical Biology, Dover, New York, 1958. doi: 10.1002/jps.3030471044. Google Scholar
[31] W. MurdochC. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, 2003. Google Scholar
[32]

P. J. PalS. SarwardiT. Saha and P. K. Mandal, Mean square stability in a modified Leslie-Gower and holling-type Ⅱ predator-prey model, J. Appl. Math. Inform., 29 (2011), 781-802. Google Scholar

[33] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0. Google Scholar
[34] E. C. Pielou, An Introduction to Mathematical Ecology, 2 edition, John Wiley & Sons, New York, 1977. doi: 10.1002/bimj.19710130308. Google Scholar
[35]

D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753. doi: 10.1137/S0036139903428719. Google Scholar

[36]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Inst. Commun., 21 (1999), 493-506. Google Scholar

[37]

C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys. Ser. B., 14 (2010), 289-306. doi: 10.3934/dcdsb.2010.14.289. Google Scholar

show all references

References:
[1]

A. A. Andronov, E. A. Leontovich, I. I. Gordon and A. G. Maier, Theory of Bifurcations of Dynamical Systems on a Plane, Israel Program for Scientific Translations, Jerusalem-London, 1973. doi: 10.1016/0022-0396(77)90136-X. Google Scholar

[2]

R. I. Bogdanov, Versal deformations of a singular point on the plan in the case of zero eigenvalues, Selecta Math. Soviet., 1 (1981), 389-421. Google Scholar

[3]

F. Brauer and A. C. Soudack, Stability regions in predator-prey systems with constant-rate prey harvesting, J. Math. Biol., 8 (1979), 55-71. doi: 10.1007/BF00280586. Google Scholar

[4]

F. D. Chen, On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay, J. Comput. Appl. Math., 180 (2005), 33-49. doi: 10.1016/j.cam.2004.10.001. Google Scholar

[5] S. N. ChowC. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, 1994. doi: 10.1017/CBO9780511665639. Google Scholar
[6] C. W. Clark, Bioeconomic Modelling and Fisheries Management, Wiley, New York, 1985. doi: 10.1016/0025-5564(87)90046-0. Google Scholar
[7]

C. W. Clark, Aggregation and fishery dynamics: A theoretical study of schooling and the purse seine tuna fisheries, Fish. Bull., 77 (1979), 317-337. Google Scholar

[8] C. W. Clark, Mathmatics Bioeconomics, The Optimal Management of Renewable Re-sources, John Wiley & Sons, Inc., New York-London-Sydney, 1976. Google Scholar
[9]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity withnilponent linear part. The cusp case of codimension 3, Ergodic Theor. Dyn. Syst., 7 (1987), 375-413. doi: 10.1017/S0143385700004119. Google Scholar

[10]

T. C. Gard, Persistence in food webs: Holling-type food chains, Math. Biosci., 49 (1980), 61-67. doi: 10.1016/0025-5564(80)90110-8. Google Scholar

[11]

Y. Gong and J. Huang, Bogdanove-Takens bifurcations in a Leslie-Gower predator-prey model with prey harvesting, Acta Math. Apple. Sinica Eng. Ser., 30 (2014), 239-244. doi: 10.1007/s10255-014-0279-x. Google Scholar

[12] J. Guckenheimer and P. Holmes, Nonlinear oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. doi: 10.1007/2F978-1-4612-1140-2. Google Scholar
[13]

R. P. GuptaM. Banerjee and P. Chandra, Bifurcation analysis and control of Leslie-Gower predator-prey model with Michaelis-Menten type prey-harvesting, Differ. Equ. Dyn. Syst., 20 (2012), 339-366. doi: 10.1007/s12591-012-0142-6. Google Scholar

[14]

R. P. GuptaP. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear harvesting, Disc. Cont. Dyna. Sys. Ser. B., 20 (2015), 423-443. doi: 10.3934/dcdsb.2015.20.423. Google Scholar

[15]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math.Anal. Appl., 398 (2013), 278-295. doi: 10.1016/j.jmaa.2012.08.057. Google Scholar

[16]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293-320. doi: 10.4039/Ent91293-5. Google Scholar

[17]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783. doi: 10.1137/S0036139993253201. Google Scholar

[18]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Disc. Cont. Dyna. Sys. Ser. B., 18 (2013), 2101-2121. doi: 10.3934/dcdsb.2013.18.2101. Google Scholar

[19]

C. JostO. Arino and R. Arditi, About deterministic extinction in ratio-dependent predator-prey model, J. Comput. Bull. Math. Biol., 61 (1999), 19-32. doi: 10.1006/bulm.1998.0072. Google Scholar

[20]

S. V. KrishnaP. D. N. Srinivasu and B. Kaymackcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584. doi: 10.1006/bulm.1997.0023. Google Scholar

[21]

Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406. doi: 10.1007/s002850050105. Google Scholar

[22] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2 edition, Springer-verlag, New York, 1998. doi: 10.1003/978-1-4757-3978-7. Google Scholar
[23]

Y. LamontagneC. Coutu and C. Rousseau, Bifurcation analysis of a predator-prey system with generalized Holling type Ⅲ functional response, J. Dynam. Diff. Equ., 20 (2008), 535-571. doi: 10.1007/s10884-008-9102-9. Google Scholar

[24]

K. Q. Lan and C. R. Zhu, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Analysis: RAW., 12 (2011), 1961-1973. doi: 10.1016/j.nonrwa.2010.12.012. Google Scholar

[25]

K. Q. Lan and C. R. Zhu, Phase portraits of predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys., 32 (2012), 901-933. doi: 10.3934/dcds.2012.32.901. Google Scholar

[26]

B. LeardC. Lewis and J. Rebaza, Dynamics of ratio-dependent predator-prey models with non-constant harvesting, Disc. Cont. Dyna. Sys. Ser. S., 1 (2008), 303-315. doi: 10.3934/dcdss.2008.1.303. Google Scholar

[27]

P. Lenzini and J. Rebaza, Non-constant predator harvesting on ratio-dependent predator-prey models, Appl. Math. Sci, 4 (2010), 791-803. Google Scholar

[28]

T. Lindstrom, Qualitative analysisnof a predator-prey system with limit cycles, J. Math. Biol., 31 (1993), 541-561. doi: 10.1007/BF00161198. Google Scholar

[29]

J. M. Lorca, E. G. Olivares and B. G. Yanez, The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, In: Mondaini, R. (ed. ) Proceedings of the International Symposium on Mathematical and Computational Biology: BIOMAT 2006, E-papers Servicos Editoriais Ltda. , R'io de Janeiro, (2007), 105–132.Google Scholar

[30] A. Lotka, Elements of Mathematical Biology, Dover, New York, 1958. doi: 10.1002/jps.3030471044. Google Scholar
[31] W. MurdochC. Briggs and R. Nisbet, Consumer-Resource Dynamics, Princeton University Press, New York, 2003. Google Scholar
[32]

P. J. PalS. SarwardiT. Saha and P. K. Mandal, Mean square stability in a modified Leslie-Gower and holling-type Ⅱ predator-prey model, J. Appl. Math. Inform., 29 (2011), 781-802. Google Scholar

[33] L. Perko, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4684-0249-0. Google Scholar
[34] E. C. Pielou, An Introduction to Mathematical Ecology, 2 edition, John Wiley & Sons, New York, 1977. doi: 10.1002/bimj.19710130308. Google Scholar
[35]

D. Xiao and L. S. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM J. Appl. Math., 65 (2005), 737-753. doi: 10.1137/S0036139903428719. Google Scholar

[36]

D. Xiao and S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Inst. Commun., 21 (1999), 493-506. Google Scholar

[37]

C. R. Zhu and K. Q. Lan, Phase portraits, Hopf bifurcations and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Disc. Cont. Dyna. Sys. Ser. B., 14 (2010), 289-306. doi: 10.3934/dcdsb.2010.14.289. Google Scholar

Figure 1.  The number of interior equilibriums of system (4)
Figure 2.  There is no interior equilibrium
Figure 3.  A unique interior equilibrium $E_2$
Figure 4.  A unique interior equilibrium $E_3$
Figure 5.  The bi-stability occurred
Figure 6.  A stable limit cycle
Figure 7.  Two limit cycles
Figure 8.  An unstable limit cycle
Figure 9.  A cusp of codimension 2
Figure 10.  An unstable limit cycle
Figure 11.  An unstable homoclinic loop
Figure 12.  A saddle and a stable focus
[1]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041

[2]

Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807

[3]

Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101

[4]

Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128

[5]

R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423

[6]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[7]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701

[8]

Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005

[9]

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75

[10]

Benjamin Leard, Catherine Lewis, Jorge Rebaza. Dynamics of ratio-dependent Predator-Prey models with nonconstant harvesting. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 303-315. doi: 10.3934/dcdss.2008.1.303

[11]

Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979

[12]

Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141

[13]

Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265

[14]

Zengji Du, Xiao Chen, Zhaosheng Feng. Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1203-1214. doi: 10.3934/dcdss.2014.7.1203

[15]

Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

[16]

Qizhen Xiao, Binxiang Dai. Heteroclinic bifurcation for a general predator-prey model with Allee effect and state feedback impulsive control strategy. Mathematical Biosciences & Engineering, 2015, 12 (5) : 1065-1081. doi: 10.3934/mbe.2015.12.1065

[17]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

[18]

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

[19]

Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021

[20]

Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (21)
  • HTML views (25)
  • Cited by (0)

Other articles
by authors

[Back to Top]